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On the $\mu $-invariant of the cyclotomic derivative of a Katz p-adic $L$-function

Part of: Discontinuous groups and automorphic forms Algebraic number theory: global fields

Published online by Cambridge University Press:  10 January 2014

Ashay A. Burungale
Ashay A. Burungale*
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA (ashayburungale@gmail.com)
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Abstract

When the branch character has root number $- 1$, the corresponding anticyclotomic Katz $p$-adic $L$-function vanishes identically. For this case, we determine the $\mu $-invariant of the cyclotomic derivative of the Katz $p$-adic $L$-function. The result proves, as an application, the non-vanishing of the anticyclotomic regulator of a self-dual CM modular form with root number $- 1$. The result also plays a crucial role in the recent work of Hsieh on the Eisenstein ideal approach to a one-sided divisibility of the CM main conjecture.

Keywords

Iwasawa invariant; Katz p-adic L-function; Hilbert modular Shimura variety; toric modular forms; anticyclotomic regulator

MSC classification

Secondary: 11F33: Congruences for modular and $p$-adic modular forms 11R23: Iwasawa theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 1 , January 2015 , pp. 131 - 148
Copyright
©Cambridge University Press 2013 

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